Russian Federal Nuclear Center – Zababakhin Institute of Applied Physics “ROSATOM” STATE CORPORATION ROSATOM An equivalent inverse source method of evaluating the weight functions for parallel-plate time-domain diffuse optical tomography Alexander B. Konovalov and Vitaly V. Vlasov Saratov Fall Meeting SFM’16, Internet Biophotonics IX Saratov, September 26-30 2016 We have a pleasure to thank the Program and Organizing Committees of SFM’16 for giving us an opportunity to present our paper in the Internet Section. Our method relates to time-domain diffuse optical tomography in parallel-plate transmission geometry.

Analytical representation of the weight functions for half-space Contents Introduction Analytical representation of the weight functions for half-space Finding the positions of the equivalent and equivalent inverse sources Numerical experiment on reconstruction of optical inhomogeneities Resolution analysis and conclusions Here you can see the contents of our paper. 2 2

Lyubimov’s perturbation model The fundamental equation is the time-resolved optical projection, is the temporal point spread function The weight functions The perturbation model by Dr. Lyubimov for reconstruction of the absorbing (Delta-mu-a) and diffusion (Delta-D) inhomogeneities is based on the fundamental equation presented here. As measurement data it uses the so-called time-resolved optical projections which are defined for a single count of the temporal point spread function registered. As you can see, the weight functions W-mu-a and W-D responsible for inhomogeneity reconstruction are expressed in terms of Creen’s function of a nonstationary diffusion equation. V.V. Lyubimov, Opt. Spectrosc. 80: 616 (1996) A.B. Konovalov et al., Proc. SPIE 8088: 80880T (2011) 3 3

Two-step approach for weight function evaluation Our approach is based on two assumptions c S D The effect of one boundary on photon migration near the other boundary is negligibly small. The distributions of photons near the source S on one boundary and near the receiver D on the other boundary are similar in nature. Our approach involves two steps We suggest a two-step approach to evaluate the weight functions for the parallel-plate transmission geometry. It is based on two assumptions: First, the effect of one boundary on photon migration near the other boundary is negligibly small; and second, the distributions of photons near the source S on one boundary and near the receiver D on the other boundary are similar in nature. These assumptions are, for example, suitable for optical mammography. In view of these assumptions we can first derive the analytical expressions for the case of semi-infinite scattering medium and then apply them to obtain the weight functions for the flat layer using the equivalent inverse source method. Derive the analytical expressions for the case of semi-infinite scattering medium Apply them to obtain the weight functions for the flat layer using the equivalent inverse source method 4 4

3D Green’s function 3D case: half-space 5 This slide presents the problem statement for the case of half-space. To determine the weight functions we must substitute Green’s function into the corresponding expressions and integrate them with respect to time. 5 5

2D Green’s function 2D case: half-plane 6 Here is the problem statement for the case of half-plane. 6 6

3D case: half-space where 7 Here are the resulting expressions for half-space. A.B. Konovalov, V.V. Vlasov, Quantum Electron. 44: 719 (2014) A.B. Konovalov, V.V. Vlasov, Proc. SPIE 9917: 99170S (2016) 7 7

2D case: half-plane where 8 Here are the results for half-plane. A.B. Konovalov, V.V. Vlasov, Quantum Electron. 44: 719 (2014) A.B. Konovalov, V.V. Vlasov, Proc. SPIE 9917: 99170S (2016) 8 8

Examples of the weight functions     The images show the weight function distributions for the parameter values we chose. 9 9

Method of the equivalent inverse source Find the position of S’ inside the flat layer and calculate the weight function for one half of the layer. Find the position of S’’ and calculate the weight function for the other half of the layer. Perform superposition of the useful halves to obtain the weight function for the full layer. To obtain the weight functions for a flat layer, we must do the following three steps. We find the position of the equivalent source S-dash inside the flat layer such that the migration of photons from point S-dash to point D of the receiver is equivalent to photon migration from point S to point D in the half of the flat layer. For this half we calculate the weight function using the formulas derived. Then we find the position of the equivalent inverse source S-double-dash and calculate the weight function for the other half of the layer. Then we perform superposition of the useful halves to obtain the weight function for the layer.

Finding the positions of the equivalent sources The position of the equivalent source S’ and the position of the equivalent inverse source S’’ are found from relations for the photon average trajectories. The coordinate can be found by solving the equation where is the error function This slide presents the technique for finding the positions of the equivalent and equivalent inverse sources. It uses the formulas for the photon average trajectories in half-space we derived earlier (see references). For the coordinate we will have A.B. Konovalov et al., Quantum Electron. 36: 1048 (2006) A.B. Konovalov et al., Optik 124: 6000 (2013)

The weight function for a 2D flat layer S Here is an example of the resulting weight function obtained after superposition. D

The reconstruction problem statement The reconstruction problem is reduced to the solution of the following system of linear algebraic equations. where is the weight matrix obtained by sampling and combining all weight functions calculated for different positions of sources and receivers is the vector of unknown local disturbances of the absorption and diffusion coefficients The slide presents the reconstruction problem statement. It is reduced to the solution of the system of linear algebraic equations you can see here. is the vector of time-resolved optical projections

Our reconstruction algorithm To solve SLAE we use an algorithm that combines the algebraic reconstruction technique (ART) with regularization by total variation (TV) norm minimization. Outline of the algorithm Step 1: Do a number of ART iterations Step 2: Process the result through TV-norm minimization Step 3: Check the stop criterion. Go to Step 1 if not satisfied Step 4: Stop The slide outlines the algorithm we use to reconstruct the optical inhomogeneities. It combines the algebraic technique with total variation regularization. The researcher interactively chooses when ART iterations should stop and TV-processing start. For the stop criterion, we use the iteration convergence rate. A.B. Konovalov, V.V. Vlasov, Proc. SPIE 9917: 99170S (2016)

The numerical phantom for study 4 mm 8 cm We conduct a numerical experiment to check the correctness of weight function calculations. Here is the first numerical phantom we chose for study.

The optical parameters we chose Tissue Fat 0.0214 0.050 10 0.0332 Tumor 0.075 0.025 13 3 0.0255 -0.0077 Fibrous 0.060 0.010 11 1 0.0302 -0.0030 The values of optical parameters we chose are suitable for optical mammography. They are presented in this table. 16 16

absorption coefficient diffusion coefficient Result of reconstruction disturbance of absorption coefficient disturbance of diffusion coefficient Here you can see the results of reconstruction for absorbing and diffusion inhomogeneities. They seem very similar because, as seen from the table of optical parameters, the relative contrasts of the ihomogeneities are quite close.

The phantoms for evaluating resolution 9; 7; 5 and 3 mm 8 cm This slide shows four phantoms we used for evaluating transverse and longitudinal resolutions.

Results of reconstruction 9 mm 7 mm 5 mm 3 mm Here are the results of reconstruction. 19 19

Resolution analysis In the centre of an 8-cm-thick object, the transverse resolution is better than 3 mm. Modulation is more than 60% The longitudinal resolution is slightly worse than the transverse one. It is close to 4 mm. To evaluate the resolution we consider the profiles of optical inhomogeneities reconstructed. On the basis of each profile the modulation transfer coefficient is determined as the relative depth of a trough between two peaks. Modulation is more than 40% 20 20

Theoretical resolution limit   The photon average trajectory (PAT) The root-mean-square deviation of photons from the PAT The spatial resolution of diffuse optical tomography depends on the thickness of the banana-shaped distribution of photon trajectories. Lyubimov (Opt. Spectroc. 86: 251 (1999)) suggested that the resolution should be evaluated from the root-mean-square deviation of photons from their average trajectory. We (Quantum Electron. 44: 239 (2014)) showed that the true formula has an additional factor ksi that depends on geometry and the reconstruction technique. In our case this factor is equal to 0.3 and 0.4 for transverse and longitudinal resolutions, respectively. The resolution limit V.V. Lyubimov, Opt. Spectrosc. 86: 251 (1999) A.B. Konovalov, V.V. Vlasov, Quantum Electron. 44: 239 (2014) 21 21

Theoretical resolution limit Here you can see how the resolution limit depends on the thickness of a flat layer. The dots show the results of the numerical experiment we have described here. 22 22

Conclusions For parallel-plate transmission geometry, the weight functions responsible for optical inhomogeneity reconstruction can be evaluated semi-analytically with the equivalent inverse source method. Our implementation of Lyubimov’s perturbation model allows a record-breaking spatial resolution to be achieved (better than 3 and 4 mm inside an 8-cm-size object in transverse and longitudinal directions, respectively). Here is the bottom line of our presentation. 23 23

Publications we refer to V.V. Lyubimov, Optical tomography of highly scattering media by using the first transmitted photons of ultrashort pulses, Opt. Spectrosc. 80(4): 616-619 (1996) V.V. Lyubimov, On the spatial resolution of optical tomography of strongly scattering media with the use of the directly passing photons, Opt. Spectrosc. 86(2): 251-252 (1999) A.B. Konovalov, V.V. Vlasov, A.G. Kalintsev, O.V. Kravtsenyuk, and V.V. Lyubimov, Time-domain diffuse optical tomography using the analytic statistical characteristics of photon trajectories, Quantum Electron. 36(11): 1048-1055 (2006) A.B. Konovalov , V.V. Vlasov, A.S. Uglov, and V.V. Lyubimov, A semi-analytical perturbation model for diffusion tomogram reconstruction from time-resolved optical projections, Proc. SPIE 8088: 80880T (2011) A.B. Konovalov, V.V. Vlasov, and V.V. Lyubimov, Statistical characteristics of photon distributions in a semi-infinite turbid medium: Analytical expressions and their application to optical tomography, Optik 124(23): 6000-6008 (2013) A.B. Konovalov and V.V. Vlasov, Theoretical limit of spatial resolution in diffuse optical tomography using a perturbation model, Quantum Electron. 44(3): 239-246 (2014) Here are the publication we refer to. A.B. Konovalov and V.V. Vlasov, Calculation of the weighting functions for the reconstruction of absorbing inhomogeneities in tissue by time-resolved optical projections, Quantum Electron. 44(8): 719-726 (2014) A.B. Konovalov and V.V. Vlasov, Total variation based reconstruction of scattering inhomogeneities in tissue from time-resolved optical projections, Proc. SPIE 9917: 99170S (2016) 24 24

The authors thank you for your time! Alexander B. Konovalov Leading Scientist a_konov@mail.vega-int.ru Vitaly V. Vlasov Head of Laboratory vitaly.vlasov.v@gmail.com Thank you for your time!